We conclude that in the trajectory-based approach the flat output
(REF ) is not only a function of \(x\) , \(u\)  and forward-shifts
of \(u\) , but also a function of backward-shifts of \(\zeta \) . Thus,
it extends the usual definition.

It is important to emphasize that the flatness
of the system (REF ) does not depend on the choice
of the functions \(g(x,u)\) . Only the representation (REF )
of the flat output may differ, while the parameterization (REF )
of \(x\)  and \(u\)  is not affected. If we would restrict ourselves to
sampled data systems with \(\operatorname{rank}(\partial _{x}f)=n\) , we could always
choose \(g(x,u)=u\) . This approach leads to a definition of flatness
as proposed in [1]}, where the flat output
is a function of \(x\) , \(u\) , and forward- and backward-shifts of \(u\) .
